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Abstract

In this paper we present a detailed theoretical study that describes the
generation of scattered light in step-index polymer optical fibers by using the
side-illumination scattering measurement technique. A detailed analysis of the
variation of the maximum angle of acceptance within the fiber has been carried
out in order to calculate the scattered light as a function of different
launching conditions. The theoretical model has been developed by using the Mie
theory for spheres in the independent-scatterer approximation.

References

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Table 1.

Principal Expressions Used in the Theoretical Calculations of the
Scattered Intensity as a Function of the Incident Angle
(αi)a

Angular
Dependence(|xQ|<ρcore)

Angular
Dependence(xQ=ρcore)

sin(θz)c

sin(θz)c(xQ,ϕx)=1−n21−(xQρcoresinϕx)2

sin(θz)c(ρcore,ϕx)=1−n2cosϕx

cosθIS

−cosαrsinθzcosϕx+sinαrcosθz

−cosαisinθzcosϕx+sinαicosθz

(sinϕPS)TM

−sinθzsinϕx(sinθzsinϕx)2+(sinαrsinθzcosϕx+cosαrcosθz)2

−sinθzsinϕx(sinθzsinϕx)2+(sinαisinθzcosϕx+cosαicosθz)2

(sinϕPS)TE

cosαrcosθz+sinαrsinθzcosϕx(sinθzsinϕx)2+(sinαrsinθzcosϕx+cosαrcosθz)2

cosαicosθz+sinαisinθzcosϕx(sinθzsinϕx)2+(sinαisinθzcosϕx+cosαicosθz)2

(E)TM

Ei=−E0(sinαix+cosαiz);Er=−E0(sinαrx+cosαrz)

Ei=-E0(sinαix+cosαiz)

(E)TE

Ei=E0y;Er=E0y

Ei=E0y

a Two particular cases have been distinguished: scatterers placed
within the core (|xQ|<ρcore) and one scatterer at the
core-cladding interface (xQ=ρcore). Ei and Er are the electric fields of the
incident and refracted beams, respectively.

Table 2.

Principal Equations Used in the Theoretical Calculations of the Scattered
Intensity from Scatterers Placed within the Core as a Function of
Incident Lateral Height (yP)a

aEi and Er are the electric fields of the
incident and refracted beams, respectively.

Table 3.

Principal Equations Used in the Theoretical Calculations of
the Scattered Intensity from One Scatterer at the Core-Cladding
Interface (ρcore=xQ2+yQ2=xP2+yP2) as a Function
of Incident Lateral Height
(yP)a

Lateral
Dependence(ρcore=xQ2+yQ2=xP2+yP2)

sin(θz)c(ϕx,yP)

1−n2cos[ϕx+arctan(yPρcore2-yP2)]

sin(θz)c(ϕx)yP=0

1−n2cosϕx

cosθIS

−sinθzcosϕx

(sinϕPS)TM

cosθz(sinθzsinϕx)2+(cosθz)2

(sinϕPS)TE

sinθzsinϕx(sinθzsinϕx)2+(cosθz)2

(E)TM

Ei=E0y

(E)TE

Ei=E0z

aEi is the electric field of the
incident beam.

Table 4.

Expressions Obtained for the Angular Distribution Function
F(θz,ϕx) in the Rayleigh Approximation for Two
Launching Conditions and Two Polarizationsa

Angular Dependence

(F(θz,ϕx))TE|xQ|<ρcore

cos2θz+sin2θzcos2ϕx

(F(θz,ϕx))TM|xQ|<ρcore

sin2θz(1−cos2ϕxsin2αr)+cos2θzsin2αr−12sin2αrsin2θzcosϕx

(F(θz,ϕx))TExQ=ρcore

cos2θz+sin2θzcos2ϕx

(F(θz,ϕx))TMxQ=ρcore

sin2θz(1−cos2ϕxsin2αi)+cos2θzsin2αi−12sin2αisin2θzcosϕx

Lateral Dependence

(F(θz,ϕx))TE|xQ|<ρcore

sin2θz

(F(θz,ϕx))TM|xQ|<ρcore

cos2θz+sin2θzcos2(ϕx+αr−αi)

(F(θz,ϕx))TExQ=ρcore

sin2θz

(F(θz,ϕx))TMxQ=ρcore

cos2θz+sin2θzcos2ϕx

a Two particular cases have been distinguished: scatterers within the
core (|xQ|<ρcore) and one scatterer at the
core-cladding interface
(xQ=ρcore).

Tables (4)

Table 1.

Principal Expressions Used in the Theoretical Calculations of the
Scattered Intensity as a Function of the Incident Angle
(αi)a

Angular
Dependence(|xQ|<ρcore)

Angular
Dependence(xQ=ρcore)

sin(θz)c

sin(θz)c(xQ,ϕx)=1−n21−(xQρcoresinϕx)2

sin(θz)c(ρcore,ϕx)=1−n2cosϕx

cosθIS

−cosαrsinθzcosϕx+sinαrcosθz

−cosαisinθzcosϕx+sinαicosθz

(sinϕPS)TM

−sinθzsinϕx(sinθzsinϕx)2+(sinαrsinθzcosϕx+cosαrcosθz)2

−sinθzsinϕx(sinθzsinϕx)2+(sinαisinθzcosϕx+cosαicosθz)2

(sinϕPS)TE

cosαrcosθz+sinαrsinθzcosϕx(sinθzsinϕx)2+(sinαrsinθzcosϕx+cosαrcosθz)2

cosαicosθz+sinαisinθzcosϕx(sinθzsinϕx)2+(sinαisinθzcosϕx+cosαicosθz)2

(E)TM

Ei=−E0(sinαix+cosαiz);Er=−E0(sinαrx+cosαrz)

Ei=-E0(sinαix+cosαiz)

(E)TE

Ei=E0y;Er=E0y

Ei=E0y

a Two particular cases have been distinguished: scatterers placed
within the core (|xQ|<ρcore) and one scatterer at the
core-cladding interface (xQ=ρcore). Ei and Er are the electric fields of the
incident and refracted beams, respectively.

Table 2.

Principal Equations Used in the Theoretical Calculations of the Scattered
Intensity from Scatterers Placed within the Core as a Function of
Incident Lateral Height (yP)a

aEi and Er are the electric fields of the
incident and refracted beams, respectively.

Table 3.

Principal Equations Used in the Theoretical Calculations of
the Scattered Intensity from One Scatterer at the Core-Cladding
Interface (ρcore=xQ2+yQ2=xP2+yP2) as a Function
of Incident Lateral Height
(yP)a

Lateral
Dependence(ρcore=xQ2+yQ2=xP2+yP2)

sin(θz)c(ϕx,yP)

1−n2cos[ϕx+arctan(yPρcore2-yP2)]

sin(θz)c(ϕx)yP=0

1−n2cosϕx

cosθIS

−sinθzcosϕx

(sinϕPS)TM

cosθz(sinθzsinϕx)2+(cosθz)2

(sinϕPS)TE

sinθzsinϕx(sinθzsinϕx)2+(cosθz)2

(E)TM

Ei=E0y

(E)TE

Ei=E0z

aEi is the electric field of the
incident beam.

Table 4.

Expressions Obtained for the Angular Distribution Function
F(θz,ϕx) in the Rayleigh Approximation for Two
Launching Conditions and Two Polarizationsa

Angular Dependence

(F(θz,ϕx))TE|xQ|<ρcore

cos2θz+sin2θzcos2ϕx

(F(θz,ϕx))TM|xQ|<ρcore

sin2θz(1−cos2ϕxsin2αr)+cos2θzsin2αr−12sin2αrsin2θzcosϕx

(F(θz,ϕx))TExQ=ρcore

cos2θz+sin2θzcos2ϕx

(F(θz,ϕx))TMxQ=ρcore

sin2θz(1−cos2ϕxsin2αi)+cos2θzsin2αi−12sin2αisin2θzcosϕx

Lateral Dependence

(F(θz,ϕx))TE|xQ|<ρcore

sin2θz

(F(θz,ϕx))TM|xQ|<ρcore

cos2θz+sin2θzcos2(ϕx+αr−αi)

(F(θz,ϕx))TExQ=ρcore

sin2θz

(F(θz,ϕx))TMxQ=ρcore

cos2θz+sin2θzcos2ϕx

a Two particular cases have been distinguished: scatterers within the
core (|xQ|<ρcore) and one scatterer at the
core-cladding interface
(xQ=ρcore).